I have a very limited scientific knowledge of this subject and a lot of anecdotal evidence.
Your sinuses are there to help equilize the air pressure between all of the openings (ears, eyes, nose, mouth) in your head. Normally, the eustachian tubes allowed fluid and air to pass between freely between your nose and ears. Tear ducts are also connected to the nose and sinuses in a similar way, but they’re very small in relation to the eustachian tubes.
If you blow your nose when there’s no congestion, the tubes will pass the air back through the ear-drum. The air will leak out from behind your ear drum once the pressure gets high enough because the ear canal stretches beyond the seal of your ear drum. (just like when you yawn and everything gets louder) Try it gently the next time you’re on a plane and your ears pop. The pressure equalizes and the pain is gone.
When you’re sick - when your sinuses are clogged - there’s no way for the eustachian tubes to function properly. You get stuffed up and can’t hear and pressure builds in your head and you get a headache. At this point, trying to blow one’s nose causes so much pressure to build up in the nasal cavity that it has to go somewhere, so the next best outlet besides the Eustachian tubes is the only outlet left - the tear ducts.
I don’t know if it was really nasal mucus coming through your eyes - it could have been, but I doubt it because the tear ducts are tiny and mucus is a pretty dense fluid. Most likely you just blew tears back through the ducts, and because you were sick and dehydrated they were a little less viscous than normal.
This happens to me a lot when I get stuffed up. In fact, when I get a really bad head cold, I sometimes can squirt tears onto my glasses if I blow through my nose hard enough. It’s a pretty interesting sight in the mirror.